Number of terms in factors of polynomial

Question

I encountered a problem:

• $ f $ is a polynomial of order $d$.
• $ g $ is a factor of $f$ of order less than $d/2$.

Prove or disprove that the number of terms of $g$ is no more than that of $f$.

(All this is in $\mathbb{Z}[x]$).

I think this is true but cannot see how to do the proof. Any hints?

Answer 1

Definitely false, for example $x^6-1$ has factors $x^2\pm x+1$.

Answer 2

False: $x^2+x+1$ is a factor of $x^9-1$.